Preheat the oven to 350° F and butter an 8- by 8-inch baking dish. In a small bowl, whisk together the flour, baking powder, and salt.,In the bowl of a stand mixer, or working with an electric hand mixer, beat together the egg yolks and 1/2 cup of sugar on medium speed until very pale and thick, 3 to 4 minutes. Meanwhile, halve the berries lengthwise. Now, beat in 1/3 cup of milk and the vanilla bean seeds. Add the flour mixture and mix until just combined.,In a large bowl with clean beaters (transfer the batter to another bowl if necessary), whip the egg whites for about 1 minute, until foamy and white. With the mixer running, slowly add the remaining 1/4 cup sugar; continue to beat until you have shiny, medium-stiff peaks, about 2 minutes. Stir about 1/4 of the egg white mixture into the batter, to loosen it up a bit, then use a rubber spatula to fold the remaining white into the batter.,______,In a small bowl, stir together all the ingredients for the milk soak. Use a butter knife to cut around the edges of the cake. Then use a toothpick or a fork to make holes all over the top of the cake. pour about 1 cup of the milk mixture all over the cake and let it soak in. Then pour another cup over the cake. Cover and chill for at least 4 hours or up to overnight.,Serve the cake with remaining mixture and more blackberries. Leftovers will keep, well-wrapped, in the fridge for up to 2 days.